Prevent texture deformation when mesh deforms
So I took Todd McIntosh's advice on messing with the mesh to try to prevent it from deforming, and I have (IMO) good results:



This image has the top and bottom of the mouth not closed at all, but the sides "fully" (only partially due to how my texture will work) closed using shapekeys.

Basically loop cut and slide a ring around the edge, then from the loop extrude and scale in.  Make sure to only scale in on the X and Y axes, or whatever two axes you want to prevent the surface from bowing in.

I have the shapekeys set up so that the sides of the mouth can't come more than halfway to the center, but the top and bottom can go all the way.  There is a little bit of intersection if you have the top mostly closed, and the sides all the way in, but for my purposes, I can work around that.

And here is a blend file with some shapekeys and vertex groups if you want to test.